Visualizing

complex functions

Enhanced phase portraits

Complex functions

\(f: \mathbb C \rightarrow \mathbb C\)

live in a 4-dimensional space

Real and Imaginary components



\(f(z)= \text{Re}(x,y) + i\, \text{Im}(x,y)\)

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Analytic landscapes

\[|f(z)|\]

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A historical analytic landscape of \(|\Gamma (z)|\) from 1909

Funktionentafeln mit Formeln und Kurven by Eugene Jahnke & Fritz Emde

Mappings

Domain coloring

Phase portraits

Domain Coloring



  1. Assign a color to every point in the complex plane.

  2. Color the domain of \(f\) by painting the location \(z\) with the color determined by the value \(f(z)\).

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The color wheel

Hue, Saturation, Brightness (HSB)

H \(\leftrightarrow\) Phase

S \(\leftrightarrow\) 1

B \(\leftrightarrow\) 1

Implementation in the computer

diagram

Implementation in the computer



  • Mathematica

  • MATLAB

  • Python

  • Java

  • C++

  • GeoGebra

  • JavaScript

    • CindyJs

    • p5.js

Basic examples

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Phase portrait

\(f(z)=z\)

\([-2,2]\times [-2,2]\)

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Phase portrait

\(f(z)=\dfrac{1}{z}\)

\([-2,2]\times [-2,2]\)

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Phase portrait

\(f(z)=\dfrac{z-1}{z^2+z+1}\)

\([-2,2]\times [-2,2]\)

Enhanced phase portraits

Elias Wegert’s work from 2012

H \(\leftrightarrow\) Phase

S \(\leftrightarrow\) 1

B \(\leftrightarrow\) \(\log |f|\) - \(\lfloor \log |f| \rfloor\)

        \(\text{phase} - \lfloor \text{phase} \rfloor\)

Wegert

Enhanced phase portrait: Level curves

Combined enhanced phase portrait

Enhanced phase portraits: Modulus

Enhanced phase portrait of \(\small f(z) =\dfrac{z-1}{z^2+z+1}\)

More examples to explore...

Roots of unity: \(z^n-1\), \(n= 2, 3, \ldots , 10\)

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Laurent series

\( f(z)=\displaystyle\sum_{n=0}^{\infty} a_n(z-z_0)^n+\displaystyle\sum_{n=1}^{\infty}\frac{b_n}{(z-z_0)^n},\)

\(R_1 <|z-z_0|< R_2. \)

\[ a_n=\frac{1}{2\pi i}\oint_C \frac{f(z)dz}{(z-z_0)^{n+1}}\quad (n=0,1,2,\ldots) \]\[ b_n=\frac{1}{2\pi i}\oint_C \frac{f(z)dz}{(z-z_0)^{-n+1}}\quad (n=1,2,\ldots) \]

annulus

Poles of order \(m\): \(\exists m \geq 1, \, b_m\neq 0\) and \(b_k=0\) for \( k>m\)

Removable singularities: If \(b_n=0, \forall n\)

Essential singularities: If \(b_k \neq 0\) for infinitely many \(k\)

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Zooming in


\( f(z)=\exp\left(\dfrac{1}{z}\right)\)

Other color schemes...

\(f(z)=0.926(z+0.073857 z^5 +0.0045458 z^9)\)

Inner function
Atomic Singular Inner Function \({\small f(z)=\displaystyle \prod_{k=1}^5\exp\left(\dfrac{z+w^k}{z-w^k}\right), w=\exp(2\pi i/5).}\)

Thank you!

Online resources:
www.dynamicmath.xyz/domain-coloring
complex-analysis.com

Contact:
j.ponce@uq.edu.au

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